How do you use the quotient rule to differentiate #1 / (1 + x²)#?

1 Answer
Feb 26, 2017

The quotient rule states that the derivative of some function that's expressed as a quotient of two other functions, such as if #f(x)=(g(x))/(h(x))#, then the derivative of #f# is given through:

#f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x))^2#

For #f(x)=1/(1+x^2)#, we see that #g(x)=1# and #h(x)=1+x^2#.

We then see that #g'(x)=0# and #h'(x)=2x#. Plugging these in gives:

#f'(x)=(0(1+x^2)-1(2x))/(1+x^2)^2#

#f'(x)=(-2x)/(1+x^2)^2#

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Footnote

If you've learned the chain rule, it's easier to do this by rewriting the function as #(1+x^2)^-1# and then seeing that the derivative is #-(1+x^2)^-2d/dx(x^2)#, identical to what we found.